Abstract Motivated by quantum quenches in spin chains, a one-dimensional toy-model of fermionic particles evolving in imaginary-time from a domain-wall initial state is solved. The main interest of this toy-model is that it exhibits the arctic circle phenomenon, namely a spatial phase separation between a critically fluctuating region and a frozen region. Large-scale correlations inside the critical region are expressed in terms of correlators in a (euclidean) two-dimensional massless Dirac field theory. It is observed that this theory is inhomogenous: the metric is position-dependent, so it is in fact a Dirac theory in curved space. The technique used to solve the toy-model is then extended to deal with the transfer matrices of other models: dimers on the honeycomb and square lattice, and the six-vertex model at the free fermion point ( Δ = 0 ). In all cases, explicit expressions are given for the long-range correlations in the critical region, as well as for the underlying Dirac action. Although the setup developed here is heavily based on fermionic observables, the results can be translated into the language of height configurations and of the gaussian free field, via bosonization. Correlations close to the phase boundary and the generic appearance of Airy processes in all these models are also briefly revisited in the appendix.
Abstract The six-vertex model with domain-wall boundary conditions is one representative of a class of two-dimensional lattice statistical mechanics models that exhibit a phase separation known as the arctic curve phenomenon. In the thermodynamic limit, the degrees of freedom are completely frozen in a region near the boundary, while they are critically fluctuating in a central region. The arctic curve is the phase boundary that separates those two regions. Critical fluctuations inside the arctic curve have been studied extensively, both in physics and in mathematics, in free models (i.e. models that map to free fermions, or equivalently to determinantal point processes). Here we study those critical fluctuations in the interacting (i.e. not free, not determinantal) six-vertex model, and provide evidence for the following two claims: (i) the critical fluctuations are given by a Gaussian free field (GFF), as in the free case, but (ii) contrarily to the free case, the GFF is inhomogeneous , meaning that its coupling constant becomes position-dependent, . The evidence is mainly based on the numerical solution of appropriate Bethe ansatz equations with an imaginary extensive twist, and on transfer matrix computations, but the second claim is also supported by the analytic calculation of and its first two derivatives in selected points. Contrarily to the usual GFF, this inhomogeneous GFF is not defined in terms of the Green’s function of the Laplacian inside the critical domain, but instead, of the Green’s function of a generalized Laplacian parametrized by the function . Surprisingly, we also find that there is a change of regime when the anisotropy parameter Δ is smaller than , with becoming singular at one point.
We study simple inhomogeneous deformations of known two-dimensional classical lattice models described by conformal field theory (CFT) at large distances. The deformations are chosen to vary slowly at lattice scales, while preserving critical behavior. Globally, we find that such systems are described by a CFT in curved space, and identify the underlying space metric. Our two examples are the Ising model and the six vertex model with domain wall boundary conditions. In the latter boundaries are also inhomogeneous, which complicates the analysis. We nevertheless solve the free case using hydrodynamics. In the presence of interactions we determine the arctic curve which separates a critical fluctuating region from an ordered (frozen) phase.
The emergence of order and geometric limit shapes in a three-dimensional (3D) Coulomb phase subject to domain wall boundary conditions (DWBC) is investigated. While the arctic circle phenomenon -- the spatial segregation of frozen and fluctuating degrees of freedom -- is well-established in the two-dimensional six-vertex model (square ice), its extension to 3D remains largely unexplored. A cubic lattice model with Ising degrees of freedom living on the edges, whose ground state manifold is governed by a divergence-free (3-in/3-out) local constraint, is considered. In the bulk, this model realizes a classical spin liquid characterized by algebraic correlations and pinch-point singularities in reciprocal space. It is demonstrated that applying DWBC partially lifts the extensive ground state degeneracy, inducing long-range magnetic order in the thermodynamic limit. Despite this ordering, it is found that the system retains a fluctuating component that exhibits the signature of a Coulomb phase. Finally, by mapping the local vertex polarization density, compelling numerical support is provided for a 3D generalization of the arctic limit shape, bridging the gap between topological constraints and emergent geometry in higher dimensions.
We revisit the study of the emptiness formation probability, the probability of forming a sequence of ` spins with the same ferromagnetic orientation in the ground-state of a quantum spin chain. We focus on two different examples, exhibiting strikingly different behavior: the XXZ and Ising chains. One has a conserved number of particles, the other does not. In the latter we show that the sequence of fixed spins can be viewed as an additional boundary in imaginary time. We then use conformal field theory (CFT) techniques to derive all universal terms in its scaling, and provide checks in free fermionic systems. These are based on numerical simulations or, when possible, mathematical results on the asymptotic behavior of Toeplitz and Toeplitz+Hankel determinants. A perturbed CFT analysis uncovers an interesting `−1 log ` correction, that also appears in the closely related spin full counting statistics. The XXZ case turns out to be more challenging. We use a simple qualitative picture in which the ferromagnetic sequence of spins freezes all degrees of freedom inside of a certain “arctic ” region, that we determine numerically. We also provide numerical evidence for the existence of universal logarithmic terms, generated by the massless field theory living outside of the arctic region. ar
Abstract Inhomogeneous quantum critical systems in one spatial dimension have been studied by using conformal field theory in static curved backgrounds. Two interesting examples are the free fermion gas in the harmonic trap and the inhomogeneous XX spin chain called rainbow chain. For conformal field theories defined on static curved spacetimes characterised by a metric which is Weyl equivalent to the flat metric, with the Weyl factor depending only on the spatial coordinate, we study the entanglement hamiltonian and the entanglement spectrum of an interval adjacent to the boundary of a segment where the same boundary condition is imposed at the endpoints. A contour function for the entanglement entropies corresponding to this configuration is also considered, being closely related to the entanglement hamiltonian. The analytic expressions obtained by considering the curved spacetime which characterises the rainbow model have been checked against numerical data for the rainbow chain, finding an excellent agreement.
In recent years the out-of-equilibrium dynamics of many-body quantum systems has attracted growing interest, both for its fundamental implications in the understanding of quantum thermalization and for potential applications in quantum information and quantum computation. In particular, quantum quenches provide a natural setting to investigate the mechanisms of correlation propagation and entanglement growth far from equilibrium. Interest in these phenomena stems not only from the possibility of testing theoretical predictions coming from emergent hydrodynamic descriptions, but also from understanding whether, and to what extent, such descriptions remain valid in the presence of spatial gradients or interfaces. Two paradigmatic frameworks in this context are the Generalized Hydrodynamics (GHD) and quasi-particle picture. The former is an effective theory devised to describe the long-time evolution after a quench in the thermodynamic limit. Within this perspective, inhomogeneous quenches are a crucial testbed for extending hydrodynamic theories to more realistic situations where, for example, scattering at an interface can occur. This is the main goal of the present thesis. The reference physical model is the one-dimensional Kitaev chain, an integrable spinless p-wave superconductor that exhibits both a topological and a trivial phase. The bilinear structure of the Hamiltonian in the fermionic creation and annihilation operators admits an exact rewriting in terms of Majorana fermions, in which the Hamiltonian remains quadratic. This representation allows a compact description of the ground-state properties via the Majorana correlation matrix (Gamma) at time t=0: the entries of Gamma completely encode the quantum information of the Gaussian ground state. The time evolution of Gamma is studied in the Heisenberg picture, where unitary dynamics correspond to an orthogonal rotation of Gamma implemented by an orthogonal matrix determined by the Kitaev Hamiltonian in the Majorana basis. Consequently, the entanglement entropy dynamics following a quench, both homogeneous and inhomogeneous, can be simulated through simple matrix products, making the approach numerically efficient even for sizable systems. The evolution code for Gamma was validated on a homogeneous quench with periodic boundary conditions: subsystem-size (l) curves exhibit the expected finite-size scaling, (Δ \propto 1/l), and numerical growth rates in the linear regime (t < v/2l), extracted by linear fits at fixed l, match the quasi-particle predictions for various values of the new chemical potential. The rate discrepancy behaves asymptotically as (Δ_{rate}} \propto 1/t_{\min}\) for large (t_{\min}\), up to oscillations from sub-extensive terms. The physical problem of interest is then an inhomogeneous quench of the chemical potential on the infinite Kitaev chain ground state with open boundaries, (μ_0 to μ_L, μ_R), producing a central interface. Initial and final Hamiltonians are free in the quasiparticle sense (ballistic propagation), but the presence of two distinct energy bands implies interface scattering. Simulations varying only μ_R (within the same trivial phase) show a nonmonotonic dependence of the entanglement growth rate, with a maximum at a particular μ_R. To explain this, an analytic ansatz for the stationary Majorana wavefunction was constructed: plane waves multiplied by a Majorana spinor, with bulk spinors obtained from the homogeneous PBC problem via the time-independent Schrödinger equation. Imposing matching at the interface yields four nontrivial equations for two unknowns (overdetermined in general). In the special cases (Δ ̸= t) two equations become trivial, the system is determined, and one obtains an explicit transmission coefficient. Combining this result with known quasi-particle formulas for two semi-infinite chains yields a hydrodynamic prediction for entanglement growth with interface scattering. Energy conservation across the interface selects the momenta that can be transmitted and thus contribute to inter-region correlations. Numerical tests for several total lengths (L) confirm the finite-size scaling (Δ_{\mathrm{e.e.}} \propto 1/L\), and show excellent agreement between numerical and theoretical growth rates for both quenches restricted to the right half and quenches applied to both halves. For the right-only case the rate discrepancy again follows (Δ_{rate} \propto 1/t_{min}\) as
tmin→∞. Possible future directions include: the study of the general inhomogeneous quench with (Δ ̸= t) via a direct extension of the analytic ansatz; the analysis of quenches starting from the topological ground state; the investigation of universal effects near the quantum critical point (μ=2t); the introduction of multiple interfaces or regions with spatially varying parameters to explore interference phenomena between quasiparticle fronts (e.g. periodic alternation (μ_1,μ_2,μ_1,μ_2,....) and, finally, the extension to interacting models where the Gaussian formalism is no longer applicable.